#128172

order	512
nilpotency class	4
isoclinism	isoclinic class of #128001
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(2,8,10,22)(3,13)(4,15)(5,18)(9,25,11,23)(12,24,27,32)(14,16)(17,31)(19,30)(21,29,28,26);(1,2,5,11,6,10,3,9)(4,8,16,23,15,22,30,25)(7,12,18,29,20,27,13,26)(14,21,17,24,19,28,31,32);(1,3,6,5)(2,9,10,11)(4,14,15,19)(7,13,20,18)(8,21,22,28)(12,26,27,29)(16,17,30,31)(23,24,25,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg,h^b=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238502, #72238505, #72238509, …
maximal normal subgroup of	#72238502, #72238505, #72238509, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35368);